1.I was now suffering from aphasia.
There is no mixing of tenses here - this is a straightforward sentence in the past simple. The word 'now' refers a previously specified time in the past.
2.Experts confirmed that a puma will not attack a human being unless it is cornered.
The confirming of this fact happened in the past, but what they confirmed is a general fact - hence the present tense 'is cornered'. By the way, the 'will' form is not a future - it also refers to a constant fact/tendency/typical behaviour.
3.I heard the forecast say the weather will be fine tomorrow.
Isn't this obvious? 'I heard' refers to something that you did in the past - this has already happened. But the future form, 'the weather will be fine', refers to 'tomorrow' - this hasn't happened yet.
4.Did you hear that he is ill?
Again, this is normal. The action of hearing the news refers to the past, but the fact that the person is ill refers to present time - this person is ill now.